Geometry of wave propagation on active deformable surfaces

Fundamental biological and biomimetic processes, from tissue morphogenesis to soft robotics, rely on the propagation of chemical and mechanical surface waves to signal and coordinate active force generation. The complex interplay between surface geometry and contraction wave dynamics remains poorly understood. Here, we couple a dispersive wave model to non-Euclidean shell mechanics to identify and characterize generic features of chemo-mechanical wave propagation on active deformable surfaces. Our theoretical framework is validated against recent data from contractile wave measurements on ascidian and starfish oocytes, producing good quantitative agreement in both cases. The theory is then applied to illustrate how geometry and preexisting discrete symmetries can be utilized to focus active elastic surface waves. Generalizing to the targeted design of active morphable materials, we conclude by demonstrating that a controlled cascade of spontaneous transitions between discrete symmetries can be induced on both the shell and the traveling wave through the careful tuning of material properties. Altogether, our results show how geometry, elasticity and chemical signaling can be harnessed to construct dynamically adaptable, autonomous mechanical surface wave guides.